LONDON — When the U.S. men’s basketball team takes to the floor Sunday at North Greenwich Arena, it will feature Kobe Bryant and LeBron James, who have won NBA championships and league MVP honors and combined to make more than $41 million last season. Two decades after the Dream Team of professionals suited up in Barcelona, this is neither a novelty nor a curiosity.

The Olympics long ago ceased being a solely amateur endeavor, but the rules on professionalism differ from sport to sport and are constantly evolving. Fans sometimes don’t know if they’re rooting for up-and-coming amateurs or the best in their fields, or something in between, and as the Summer Games draw to close Sunday, the athletes departing London on Monday are heading toward radically varied lifestyles.

Olympic swimming, for instance, allows pros. So does track and field. Boxing does not. Less-prominent sports have no rules against professionals, but it can be difficult for those professionals to support themselves as they train.

On Thursday night, 80,203 fans filled historic Wembley Stadium here to watch the U.S. beat Japan for the gold medal in women’s soccer, the conclusion of a tournament that included all of the best players in the world. On Saturday, another sold-out crowd came to watch Brazil face Mexico at Wembley, the conclusion of a tournament which the best players watched from afar, because each roster consisted of players age 23 or under, save for three exceptions who could be of any age.

“It’d be nice to have the best players here,” said Tom Cleverley, a midfielder for Britain’s entry in the men’s soccer tournament. “But we have other things coming up that are more important to more people,” a reference to the English Premier League, in which Cleverley is a member of Manchester United.

Different sets of rules

The Olympics, it turns out, have only so much control over Olympic sports. Each sport’s international governing body — the International Federation of Association Football for soccer, the International Basketball Federation for basketball, etc. — decides who can compete in its sport at the Olympics and in other international competitions.

USA Swimming, which has fielded perhaps the most successful American program dating from the 2004 Games in Athens, supports everyone from 22-time medalist Michael Phelps to 15-year-old Katie Ledecky, the Bethesda high schooler who won her first gold here at her first Games. But even within that structure — in which athletes can and do receive endorsement contracts — there are struggles.

Missy Franklin, a 17-year-old high school student, could command hundreds of thousands of dollars after she won five medals here. But Franklin wants to swim collegiately, and the NCAA — yet another organization that has its fingers on the Olympics, albeit from afar — won’t allow pros to compete.

Saturday, Franklin said her experience in London has left her torn on how to proceed.

“I’ve been able to see the benefits [of] how people get these sponsorships and what it’s like for them and how much fun they’re having, and kind of wanting that and wanting to be part of it, and having it be so hard to turn it down,” she said. “But it’s also helped me in being part of such an incredible team . . . being on the closest team I’ve ever been on and knowing that that’s exactly what college is going to be like.”

Franklin believes, though, she has nothing but good choices. In sports less internationally prominent — and lucrative — than basketball or soccer or swimming, financial constraints can have an enormous impact on athletes’ ability to continue their sport.

“People don’t realize what it costs the family,” said Christine Lofgren, whose daughter Esther won gold as a member of the U.S. women’s eight rowing team.

The United States Olympic Committee offers incentives for performance: $25,000 for a gold medal, $15,000 for silver and $10,000 for bronze. So Franklin, in trying to maintain her eventual college eligibility, is leaving $110,000 on the table, the money she could take for her four golds and one bronze in London.

The USOC provides support in other ways as well. In beach volleyball, for instance, the top two teams receive health insurance and a modest stipend during their season, and the top four teams receive help with travel costs. But athletes say it’s hardly enough to pursue an Olympic title and train for four years, and they have occasionally gotten creative. Nick Symmonds, a U.S. middle distance runner, auctioned off advertising space on his body this spring, earning $11,000 in return for a temporary tattoo.

“The rest is out of pocket,” said Jennifer Kessy, who won silver here in women’s beach volleyball, alongside April Ross. “We both started at the bottom with no funding, and we’ve seen it get bigger and bigger. Hopefully these medals will help future athletes for some funding.”

Which way to go?

In the few sports that prohibit true professionals, the stipend usually only covers a portion of training costs. Claressa Shields, a 17-year old from Flint, Mich., who became the first American woman to win gold in boxing, receives a monthly stipend of $1,000 and will weigh her financial opportunities as she decides whether to return for the 2016 Games or focus on a professional career.

“If USA Boxing wants to keep her around, they want to help her out, give her a little bit more,” said her coach, Jason Crutchfield, with whom Fields lives for much of the year. “Give her a little more than $1,000 a month, then she’ll stay in the ring. That’s not enough. Her being a kid, she thinks $1,000 is enough to her. She don’t pay no bills.”

Boxing, too, is preparing to move the way of basketball, albeit 20 years later. The International Boxing Association plans to introduce pro fighters in the 2016 Games. The organization has recently launched World Series of Boxing, a professional outlet that it hopes will help grow the organization. To encourage participation, more than 50 fighters who participate could be eligible to compete in Rio.

Basketball is considering a move in the opposite direction. As Bryant, James and their teammates try to win the Americans’ fifth gold medal in the six Summer Games since the best players in the world were allowed to compete in the Olympics — regardless of salary or age — there is a movement, led by NBA Commissioner David Stern, to dial back yet again. Stern’s idea: Make the Olympic tournament a 23-and-under event, better to preserve older bodies for the long NBA season and make basketball’s biannual world championships more meaningful.

Larry Probst, the chairman of the USOC, was asked Saturday about basketball’s future at the Games. He could only offer a personal opinion, because the decision will ultimately be made by NBA and IBF officials — not those directly involved in the Olympic movement.

“I personally would like to see the best players in the world,” Probst said. “And if they happen to be 35 or 37 or 27 or 19, I’d like to see us field the very best team that we could put on the court.”

Depending on the sport, whether it be throughout the London Games or those in four years in Rio de Janeiro, the very best team may or may not be present, and it could change the next time around.

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